Find the ratio in which the line ` 2x+ 3y -5=0 ` divides the line segment joining the points (8,-9) and (2,1) Also. Find the co-ordinates of the point of divisions

Answer

To solve the problem of finding the ratio in which the line \(2x + 3y - 5 = 0\) divides the line segment joining the points \((8, -9)\) and \((2, 1)\), we will follow these steps:

### Step 1: Identify the points and the line equation
Let the points be:
- \(A(8, -9)\)
- \(B(2, 1)\)

The line equation is given as:
\[ 2x + 3y - 5 = 0 \]

### Step 2: Assume the point of division
Let the point \(K\) divide the segment \(AB\) in the ratio \( \lambda : 1 \). According to the section formula, the coordinates of point \(K\) can be expressed as:
\[ K\left(\frac{2\lambda + 8}{\lambda + 1}, \frac{\lambda - 9}{\lambda + 1}\right) \]

### Step 3: Substitute the coordinates of \(K\) into the line equation
Since point \(K\) lies on the line \(2x + 3y - 5 = 0\), we substitute the coordinates of \(K\) into the equation:
\[ 2\left(\frac{2\lambda + 8}{\lambda + 1}\right) + 3\left(\frac{\lambda - 9}{\lambda + 1}\right) - 5 = 0 \]

### Step 4: Simplify the equation
Multiply through by \((\lambda + 1)\) to eliminate the denominator:
\[ 2(2\lambda + 8) + 3(\lambda - 9) - 5(\lambda + 1) = 0 \]
Expanding this gives:
\[ 4\lambda + 16 + 3\lambda - 27 - 5\lambda - 5 = 0 \]

### Step 5: Combine like terms
Combine the terms:
\[ (4\lambda + 3\lambda - 5\lambda) + (16 - 27 - 5) = 0 \]
This simplifies to:
\[ 2\lambda - 16 = 0 \]

### Step 6: Solve for \(\lambda\)
Solving for \(\lambda\):
\[ 2\lambda = 16 \implies \lambda = 8 \]

### Step 7: Determine the ratio
The ratio in which the line divides the segment \(AB\) is:
\[ \lambda : 1 = 8 : 1 \]

### Step 8: Find the coordinates of the point of division
Now we substitute \(\lambda = 8\) back into the coordinates of point \(K\):
\[ K\left(\frac{2(8) + 8}{8 + 1}, \frac{8 - 9}{8 + 1}\right) = K\left(\frac{16 + 8}{9}, \frac{-1}{9}\right) = K\left(\frac{24}{9}, \frac{-1}{9}\right) = K\left(\frac{8}{3}, -\frac{1}{9}\right) \]

### Final Result
Thus, the ratio in which the line divides the segment is \(8:1\) and the coordinates of the point of division are:
\[ K\left(\frac{8}{3}, -\frac{1}{9}\right) \]